hereditary relatively;injective subquivers and equivalence modulo preprojectives
TRANSCRIPT
This article was downloaded by: [Universitaets und Landesbibliothek]On: 03 September 2013, At: 17:47Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK
Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20
Hereditary relatively;injective subquivers andequivalence modulo preprojectivesHéctor A. Merklen aa Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP,BrasilPublished online: 27 Jun 2007.
To cite this article: Hctor A. Merklen (1990) Hereditary relatively;injective subquivers and equivalence modulopreprojectives, Communications in Algebra, 18:9, 3145-3181, DOI: 10.1080/00927879008824065
To link to this article: http://dx.doi.org/10.1080/00927879008824065
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”)contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy, completeness, or suitabilityfor any purpose of the Content. Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy ofthe Content should not be relied upon and should be independently verified with primary sources ofinformation. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial orsystematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution inany form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
COMMUNICATIONS IN ALGEBRA, 1 8 ( 9 ) , 3145-3181 (1990)
HEREDITARY RELATIVELY INJECTIVE SUBQUIVERS
AND EQUIVALENCE MODULO PREPROJECTIVES
~ e c t o r A, Merklen I n s t i t u t o de ~ a t e m a t i c a e ~ s t a t I s t i c a
Un ive r s idade de S ~ O Paulo S ~ O P a u l o , SP, B r a s i l
An a r t i n a l g e b r a A i s s a i d t o be e q u i v a l e n t t o a n
a l g e b r a A ' modulo p r e p r o j e c t i v e s up t o t h e l e v e l n
if t h e c a t e g o r i e s mod A/add p n ( A ) and mod A1 /add ~ ( A ' I
a r e e q u i v a l e n t , Necessary and s u f f i c i e n t c o n d i t i o n s f o r
t h i s a r e g iven i n t h e c a s e when A ' i s h e r e d i t a r y o f
i n f i n i t e r e p r e s e n t a t i o n t y p e , assuming c e r t a i n p r o p e r -
t i e s f o r t h e p r e p r o j e c t i v e modules i n g n ( n ) .
1. I n t r o d u c t i o n and s t a t e m e n t o f t h e r e s u l t .
I n t h i s a r t i c l e we c o n t i n u e t h e r e s e a r c h begun i n
[ 4 ] . Given an a r t i n a l g e b r a A and a n a t u r a l number n ,
(Pm(A)), i s t h e p r e p r o j e c t i v e p a r t i t i o n o f i n d A ( s e e
[ 31 ) and p n ( h ) d e n o t e s t h e subca t ego ry d e f i n e d by t h e
Copyright @ 1990 by Marcel Dekker, Inc.
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
modules i n { !,CAI / 0 9 m < n ) . By d e f i n i t i o n , A
i s s a i d t o be e q u i v a l e n t t o a n o t h e r a r t i n a l g e b r a A '
modulo p r e p r o j e c t i v e s up t o t h e l e v e l n i f mod A
a d d P " ( A )
and " a r e e q u i v a l e n t c a t e g o r i e s . The c;se add p n ( ~ ' - I
n = 0 i s t h e c a s e o f t h e u s u a l s t a b l e e q u i v a l e n c e ,
The prob lem of c h a r a c t e r i z i n g s t a b l e e q u i v a l e n c e
w i t h a h e r e d i t a r y a l g e b r a was s t a t e d a n d s o l v e d by Aus-
l a n d e r and R e i t e n i n [ l ] , 1972. The c o n d i t i o n s o b t a i n e d
t h e r e were t h a t , f i r s t , e v e r y indecomposable non p r o j e c -
t i v e submodule o f a p r o j e c t i v e module i s s i m p l e a n d , s e -
c o n d l y , t h a t t h o s e s i m p l e a r e q u o t i e n t s o f i n j e c t i v e
modules . E q u i v a l e n t l y , e v e r y indecomposable q u o t i e n t o f
a n indecomposable i n j e c t i v e module i s i n j e c t i v e o r s i m -
p l e a n d , i n t h e l a t t e r c a s e , i s c o n t a i n e d i n a p r o j e c -
t i v e module. T h i s work was ba sed main ly on t h e t h e o r y o f
c a t e g o r i e s o f f u n c t o r s . The g e n e r a l i z a t i o n o f t h i s q u e s t i o n a p p e a r e d i n a
n a t u r a l way a f t e r t h e d i s c o v e r y o f t h e p r e p r o j e c t i v e mo-
d u l e s i n [ 3 1 , 1980. I n [ 4 ] , 1986 , we o b t a i n e d some ne-
c e s s a r y and some s u f f i c i e n t c o n d i t i o n s f o r A t o be
e q u i v a l e n t t o a h e r e d i t a r y a l g e b r a A ' modulo p r e p r o -
j e c t i v e s up t o t h e l e v e l n , Our s t u d y was o b v i o u s l y li-
mi t ed t o t h e i n f i n i t e r e p r e s e n t a t i o n t y p e c a s e and was
based main ly on t h e e x i s t e n c e and ~ r o ~ k r t i e s o f Auslan- Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3147
de r -Re i t en s equences . Let us r e c a l l some te rminology and
some f a c t s from [ 41 . We deno te by 1 ( r e s p . y ' ) t h e c a t e g o r y add p n ( ~ )
( r e s p . add E " ( A ' 1) . I f t h e above mentioned e q u i v a l e n c e
e x i s t s , t h e A-modules which co r r e spond t o i n j e c t i v e A ' -
modules a r e t h e ! - i n j ec t ive A-modules, whose c a t e g o r y
o f indecomposables i s denoted by I. i s t h e p a r t o f -
indVA c o n s i s t i n g o f i n j e c t i v e modules, o f modules I - -
such t h a t TrD I i s i n and of modules I such t h a t - every a r row o f t h e Aus lander -Rei ten q u i v e r T,, c o r r e s -
ponding t o an i r r e d u c i b l e map o f t h e form I -+ X has
X i n y. The V - i n j e c t i v e s - o f t h e l a t t e r k ind a r e s a i d . -
t o be y-simple, -
I n g e n e r a l , we u s e t h e same n o t a t i o n f o r a s u b c a t e -
gory o f indecomposable modules and f o r t h e co r r e spond ing
subqu ive r of t h e Aus lander -Rei ten q u i v e r , This p e r m i t s
s imp le s t a t e m e n t s f o r t h e two p r o p e r t i e s o f I - which
a r e c r u c i a l t o o u r q u e s t i o n .
1) I i s open t o t h e r i g h t i n r A \ p n ( h ) , -
2 ) I - h a s no o r i e n t e d c y c l e s and c o n t a i n s t h e left- - boundary o f -
Here , t h e f i r s t p r o p e r t y means t h a t eve ry a r row 1 4 X
w i t h I i n and X n o t i n has n e c e s s a r i l y X - -
i n I . - The l a s t p a r t o f t h e second p r o p e r t y means t h a t -
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
3148 MERKL EN
eve ry a r row X + P w i t h P i n p n ( h ) - and X n o t i n
h a s n e c e s s a r i l y X i n 5 ,
When 1) and 2) a r e s a t i s f i e d , Comod(add x / y ) - - ( s e e
s e c t i o n 3 f o r a d e f i n i t i o n ) i s a c a t e g o r y of t h e form
mod A:';, where A?: i s a h e r e d i t a r y a l g e b r a . For t h i s f
r e a s o n . we w i l l a p p l y t o o b j e c t s I -a J o f Comod(
add I/!) t h e u s u a l t e rmino logy and t h e w e l l known pro- -
p e r t i e s o f modules o f a h e r e d i t a r y a l g e b r a . I n c a s e t h a t
A i s e q u i v a l e n t t o a h e r e d i t a r y a l g e b r a A' modulo
p r e p r o j e c t i v e s u p t o t h e l e v e l n , and when A h a s no
r i n g components of f i n i t e r e p r e s e n t a t i o n t y p e , it was
shown i n I 4 1 t h a t A ' i s Mor i t a e q u i v a l e n t t o A?:.
A l l t h i s s u g g e s t s a s a n a t u r a l c o n t i n u a t i o n o f t h i s work
t o pe r fo rm a c l o s e r s t u d y o f c a t e g o r i e s o f t h e form Co-
rnod(add S / V ) , - - and t h i s i s what we b e g i n i n t h e p r e s e n t
p a p e r ,
As i t t u r n s o u t , t h e r e i s a c l o s e r e l a t i o n between
Comod(add I) - and Como?(add I / y ) when a l l a r rows I + J - -
o f co r r e spond t o i r r e d u c i b l e maps which a r e epirnor-
phisms ( i n t h i s c a s e we w i l l s ay t h a t A h a s t h e *-pro-
p e r t y ) . On t h e o t h e r hand, we have n o t been a b l e t o f i n d
a n example where A i s e q u i v a l e n t t o A ' ( h e r e d i t a r y ,
indecomposable , o f i n f i n i t e r e p r e s e n t a t i o n t y p e ) modulo
p r e p r o j e c t i v e s up t o t h e l e v e l n and where A f a i l s
t o s a t i s f y t h e * - p r o p e r t y .
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3149
I t s h o u l d be n o t e d a l s o t h a t t h e " -p rope r ty i s a lways
t r u e i n t h e c l a s s i c a l c a s e o f n = 0 ,
We d e f i n e I t o be h e r e d i t a r y i f t h e p r o p e r t i e s 1) - and 2 ) above and t h e * - p r o p e r t y a r e s a t i s f i e d ( c f . s e c -
t i o n 2 , Def .l) a If I i s h e r e d i t a r y , mod A/! i s e q u i - - - v a l e n t t o a c a t e g o r y o f t h e form mod A:':/!>':. I f , f u r - - t h e rm ore , t h e r e a r e no i n j e c t i v e modules i n Y, t h e n A
i s e q u i v a l e n t t o A" modulo p r e p r o j e c t i v e s up t o t h e
l e v e l n. We can a l s o c h a r a c t e r i z e t h i s s i t u a t i o n by
means o f c o n d i t i o n s which a r e ve ry s i m i l a r t o t h e Aus-
l a n d e r - R e i t e n c o n d i t i o n s of f 1 1 .
THEOREM. L e t A b e an a r t i n a l g e b r a s u c h t h a t t h e r e are
no i n j e c t i v e modules i n P"(A) and l e t I m ( f o ) d e n o t e - t h e f u l l s u b c a t e g o r y o f i q d A d e f i n e d by t h e indecom-
p o s a b l e q u o t i e n t s o f i n j e c t i v e .A-modules. Then t h e f o l -
l owing s t a t e m e n t s a r e e q u i v a l e n t ,
1) I - i s h e r e d i t a r y .
2 1 The f o l l o w i n g two c q n d i t i o n s a r e s a t i s f i e d .
f 2al If I --+ X i s a A-homomorphism w i t h I i n
a n d X i ndecomposab l e , t h e n f # 0 i f - - - a n d o n l y i f f i s a n epimorphism,
2b) Im(So) c o n s i s t s o f t h e indecomposable i n j e c -
t i v e modules and o f t h e indecomposable compo-
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
MERKLEN
n e n t s o f modules o f t h e form r a d P , w i t h P
i n y, which a r e n o t i n y . - - -
Let us assume f u r t h e r t h a t A ha s t h e f o l l o w i n g p rope r -
L Y " - f
I f M --+ N i s a A-epimorphism w i t h M i n modVA, - - - t h e n f # 0, - -
I n t h i s c a s e , i f A i s e q u i v a l e n t modulo p r e p r o j e c t i v e s
up t o t h e l e v e l n t o a h e r e d i t a r y a l g e b r a w i t h no r i n g
components o f f i n i t e r e p r e s e n t a t i o n t y p e , t h e n I - - i s
h e r e d i t a r y , Converse ly , i f - i s h e r e d i t a r y , , t h e n A
i s e q u i v a l e n t modulo p r e p r o j e c t i v e s up t o t h e l e v e l n
t o a h e r e d i t a r y a l g e b r a and A ha s t h e above s t a t e d
p r o p e r t y , F i n a l l y , i t i s always t r u e t h a t i f I - - i s he-
r e d i t a r y t h e n I - = Im( :o ) .
The pape r i s o rgan ized a s f o l l o w s . I n s e c t i o n 2 , un-
d e r t h e assumpt ion t h a t A i s e q u i v a l e n t modulo p rep ro -
j e c t i v e s up t o t h e l e v e l n t o a h e r e d i t a r y a l g e b r a A '
w i t h no r i n g components o f f i n i t e r e p r e s e n t a t i o n t y p e ,
and s u p o s i n g t h a t A s a t i s f i e s t h e p r o p e r t y s t a t e d i n
t h e Theorem, i t i s proved t h a t E - i s h e r e d i t a r y ( s e e
Prop. 2 . 2 ) , Some o t h e r p r o p e r t i e s a r e o b t a i n e d which a r e
imp l i ed by t h e f a c t t h a t - i s h e r e d i t a r y , and t h e no-
t i o n o f 2 - r e p r e s e n t a t i v e , - f o r a morphism o f add I / y , - - Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVEE 3151
i s i n t r o d u c e d , The p r e p a r a t i o n 1-emma (P rop . 2,6) p l a y s
an i m p o r t a n t r o l e i n p r o v i n g t h e c l o s e c o n n e c t i o n b e t -
ween Comod(add & ) and Comod(add I/Y) when I i s - - - h e r e d i t a r y . In s e c t i o n 3 , t h i s c o n n e c t i o n i s e s t a b l i s h e d
i n Prop. 3 . 3 . The s i g n i f i c a n c e o f t h e f a c t t h a t A h a s
no i n j e c t i v e modules i n 1 i s a l s o examined. I n Prop .
3 . 7 it i s shown t h a t i t i m p l i e s t h a t i s e q u a l t o
I m ( I o ) , F i n a l l y , s e c t i o n 4 i s devo t ed t o t h e p roo f o f
t h e main theorem and t o p r e s e n t a n example t o show t h a t ,
even in t h e c a s e of o u r theorem, t h e r e a r e i n s t a n c e s of
e q u i v a l e n c e modulo p r e p r o j e c t i v e s which do n o t be long
t o t h e c l a s s i c a l c a s e o f s t a b l e e q u i v a l e n c e .
2, I n t r o d u c t i o n of t h e h e r e d i t a r y ! - i n j e c t i v e q u i v e r s . ---'
We keep t h e n o t a t i o n s of [ 41, some o f which have
been a l r e a d y r e c a l l e d i n t h e i n t r o d u c t i o n . Through most
o f t h i s s e c t i o n we w i l l assume t h a t A i s e q u i v a l e n t ,
modulo p r e p r o j e c t i v e s up t o t h e l e v e l n , t o a h e r e d i -
t a r y a l g e b r a A' w i t h no r i n g components of f i n i t e r e -
p r e s e n t a t i o n t y p e , Thus, t h e ! - i n j e c t i v e q u i v e r & i s
open t o t h e r i g h t i n r A \ p n ( n ) , ha s no o r i e n t e d c y c l e s
and c o n t a i n s t h e l e f t boundary of y . - I t h a s been
shown i n [ 4 ] t h a t eve ry i r r e d u c i b l e map I -t J , c o r - Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
MERKLEN 3152
r e spond ing t o an arrow of 2 , - i s such t h a t f i s a n
epimorphism, On t h e o t h e r hand , mod A ' , which i s ob-
v i o u s l y e q u i v a l e n t t o C o m ~ d ( a d d ( ~ ~ ( A ' ) ) v i a t h e k e r -
n e l f u n c t o r , i s a l s o e q u i v a l e n t t o Comod(add I/y). - - To
s i m p l i f y o u r w r i t i n g , i f we want t o r e f e r t o t h i s s i t u a -
t i o n , we w i l l merely s ay t h a t A i s e q u i v a l e n t t o A' ,
We beg in by f i n d i n g o t h e r ways o f s t a t i n g t h a t A
has t h e " -p rope r ty (assuming t h a t it i s e q u i v a l e n t t o
A , Let us r e c a l l t h a t , f o r a A-module P, T (P ) de-
n o t e s t h e t r a c e o f indVA i n P , i , e . t h e sum o f a l l - -
images o f morphisms M -+ P f o r M i n modyA. - -
P r o p o s i t i o n 2 . lo Let A be e q u i v a l e n t t o A . The f o l -
lowing a r e e q u i v a l e n t ,
(1) I f P i s i n y, T(P) i s c o n t a i n e d i n t h e - - r a d i c a l o f P a
f (2) Every i r r e d u c i b l e map I + J w i t h I i n
2 and J i n modVA i s a n epimorphism. - - - ( 3 ) Every non z e r o epimorphism M + N w i t h M
i n modVA i s non z e r o modulo - - - - ( 4 ) Every i r r e d u c i b l e map I -+ P w i t h I i n
J and P i n f ( ~ ) h a s i t s image i n r a d P . -
PROOF, We show t h e i m p l i c a t i o n s (1) = > ( 3 ) = > ( 2 ) => Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3153
use Prop , 3 , s of [ 41 1. (2) => ( 4 ) , Let us c o n s i d e r t h e
Aus lander -Rei ten sequence beg inn ing a t I .
Here, P ( r e s p . J) d e n o t e s t h e !-part ( r e s p a t h e mod A- !
p a r t ) o f t h e middle t e rm. We say t h a t i I r e s p . f ) i s
t h e !-component ( r e s p . t h e I-component) of t h e minimal
l e f t a lmos t s p l i t map going o u t from I . S i n c e when P i s
p r o j e c t i v e i ( I ) i s c o n t a i n e d i n r a d P , w e c a n proceed b y
i n d u c t i o n on t h e minimal l e n g t h o f a p a t h o f 5 l i n k i n g - a s o u r c e t o I , by assuming t h a t i ( I ) i s c o n t a i n e d i n
r a d P. Then, J = I = p i i s c o n t a i n e d i n r a d
Q. ( 4 ) => ( l ) , It i s e a s y t o s e e , by i n d u c t i o n , t h a t a
!-envelope o f I , f o r I i n I , i s o f t h e form
where each i s o b t a i n e d i n t h e f o l l o w i n g way. For each
p a t h T o f 2 , s t a r t i n g a t I and end ing up a t some
e lement K o f I t h a t i s n o t i n j e c t i v e , we choose
a map f T , a composi te o f i r r e d u c i b l e maps one f o r e a c h
ar row o f T I f iT i s t h e !-component o f t h e minimal - l e f t a lmos t s p l i t map going o u t from K , t h e n @ i s
t h e composi te i f Th i s shows t h a t , f o r eve ry I i n
2, t h e image o f t h e !-envelope o f I i s c o n t a i n e d i n - -
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
MERKLEN 3154
t h e r a d i c a l o f t h e c o r r e s p o n d i n g module o f Y. S i n c e ,
by [ 4 1 , T (P ) i s i n a d d z, - ( 1 ) f o l l o w s e a s i l y .
By Theorem 1 o f [ 4 ] and P rop . 2 , 1 , we have t h e
f o l l o w i n g r e s u l t .
P r o p o s i t i o n 2 . 2 . Le t A b e e q u i v a l e n t t o A ' a nd l e t
us assume t h a t A h a s t h e " -p rope r ty and t h a t no i n j e c -
t i v e A-module i s i n y, - Then t h e q u i v e r I - of the- - i n j e c t i v e A-modules s a t i s f i e s t h e f o l l o w i n g p r o p e r t i e s .
1 ) I - i s open t o t h e r i g h t i n ~ * \ E " ( A ) , "aS
no o r i e n t e d c y c l e s and c o n t a i n s t h e l e f t
boundary o f y . -
2 ) Every i r r e d u c i b l e map I -+ J w i t h I i n I - and J i n modVA is a n ep imorphism,
- -
D e f i n i t i o n 1. The ! - i n j e c t i v e q u i v e r I i s s a i d t o b e - - h e r e d i t a r y when t he -_p rope r t i e s 1) and 2 ) o f Prop . 2.2
a r e s a t i s f i e d , I t i s e a s y t o s e e t h a t t h e h y p o t h e s i s o f
Prop. 2 . 1 may b e s u b s t i t u t e d by p r o p e r t y 1) o f P rop .2 .2 .
Hence, i n t h i s d e f i n i t i o n , one may s u b s t i t u t e any o f
t h e c o n d i t i o n s ( 1 1 , ( 3 ) , ( 4 ) o f Prop . 2 . 1 f o r p r o p e r t y
2 ) (which i s t h e same a s ( 2 ) o f P r o p , 2 . 1 ) .
The f o l l o w i n g lemma i m p l i e s t h a t , i f i s h e r e d i -
t a r y , t h e 1-component - o f t h e minimal a l m o s t s p l i t map
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3155
going o u t from a n indecomposable i n I i s computed a s
i f t h i s indecomposable were a n i n j e c t i v e module.
Lemma. Let A be an a r t i n a l g e b r a and M a n indecompo-
s a b l e A-module such t h a t t h e minimal l e f t a lmos t s p l i t --
map g o i n g o u t frpm- M has t h e f o l l a w i n g form.
i ,77 L
M / i ( M ) i s c o n t a i n e d i n r a d L
\ 63 where
f \N f(M) = N
a n - isomorphism u
such t h a t f = ug, where g i s t h e n a t u r a l epimorphism
from M o n t o M/soc M .
PROOF. Le t S be a s i m p l e submodule o f M and g t h e n a t -
u r a l map M -+ M/S. Then g f a c t o r s i n t h e form g = f ' f
+ i ' i , from which it f o l l o w s t h a t f ' f ( M ) = f l ( N ) = M/S.
T h i s i m p l i e s , c l e a r l y , t h a t f ' i s an isomorphism and ,
s i n c e f i s f i x e d , S = k e r f i s e q u a l t o s o c M .
I n a l l t h e r e s t o f t h i s s e c t i o n we w i l l suppose t h a t
I i s h e r e d i t a r y . S i n c e add I/! may be though t a s t h e - - - c a t e g o r y o f t h e i n j e c t i v e modules o f a h e r e d i t a r y a l g e -
f b r a , e v e r y morphism I J i n t h i s c a t e g o r y h a s an e s -
s e n t i a l l y unique f a c t o r i z a t i o n = & where h - i s an
epimorphism and where g i s s p l i t monic. I n p a r t i c u l a r ,
i f J i s indecomposable , e i t h e r f i s an epimorphism
o r f = O ,
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
3156 MERKLEN
We w i l l s a y t h a t f i s an & - r e p r e s e n t a t i v e - o f f
i f f admi t s a f a c t o r i z a t i o n f = gh where h i s an
epimorphism and g i s s p l i t monic. I t i s c l e a r t h a t i f
$ i s a morphism o f t h e c a t e g o r y add I/! - t h e n $ has
an I - r e p r e s e n t a t i v e f . I t i s e a s y t o s e e a l s o t h a t , - g iven two f a c t o r i z a t i o n s f = gh = g ' h ' where h , h '
a r e epimorphisms and g , g ' a r e s p l i t monic, t h e r e i s
- 1 an isomorphism u such t h a t h ' = uh and g ' = gu
f P r o p o s i t i o n 2 . 3 . Le t I be h e r e d i t a r y and I + J - a -
morphism w i t h I i n add 2. Then, f i s a n epimor- - phism i f and o n l y i f f i s an epimorphism. If t h i s i s
t h e c a s e , J i s a l s o i n add I , -
PROOF. We o b s e r v e f i r s t t h a t , s i n c e Prop . 2 . 1 , ( 3 ) i s
s a t i s f i e d when 4 i s h e r e d i t a r y , A-epimorphisms go ing - from modules o f modVA a r e n o t z e r o modulo y. I t f o l -
- - lows t h a t i f f i s a n epimorphism, s i n c e cok f = 0,
cok f = 0 and f i s a n epimorphism. For t h e c o n v e r s e ,
we proceed by i n d u c t i o n on t h e l e n g t h o f I , % ( I ) . I t is
c l e a r t h a t we can assume t h a t I i s o f t h e form I1 @
I2 ( i t may happen t h a t I2 = 0 1 , w i t h I1 indecompo-
s a b l e and f l = f l I l i s n o t a n isomorphism. I f f l i s
1 g s p l i t monic, f h a s t h e form f = ( O h):Il@ I2 ---t I1 @
J2 a n d , i f f is a n epimorphism, h i s a n epimorphism
t o o , By t h e i n d u c t i o n h y p o t h e s i s , 5 i s an epimorphism
and , t h e r e f o r e , f i s a n e p i m o r p h i ~ r n , I f f l i s n o t
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROSECTIVES 31 51
s p l i t monic, f f a c t o r s through a map of t h e form ( i l
0 Il @ I2 + t Ii @ 12, where tl is t h e minimal
l e f t almost s p l i t map going o u t from 11, so t h a t we
have, say , f = f ' t . By t h e induc t ion hypo thes i s , f'
i s an epimorphism and, hence f i s an epimorphism.a -
We have mentioned a l r e a d y t h a t i f I i s h e r e d i t a r y - t h e Asmodules I i n add T / y may be thought a s t h e - - i n j e c t i v e modules of some h e r e d i t a r y a lgebra A*. I n
what fo l lows , 1 ( I ) w i l l denote t h e l eng th o f I a s an - i n j e c t i v e A*-module and, i f - f : I ---t J i s a morphism
of add x/y, k e r f w i l l denote i t s k e r n e l a s a morphism - - - of A*-modules ,
Propos i t ion 2 . 4 . Let be h e r e d i t a r y . I f I i s i n - add H, - then L(I) = b(I), -
PROOF, We proceed by induc t ion on l ( 1 ) . By Prop. 2 . 3
and by t h e lemma above, I i s simple i f and only i f it
i s g-simple and i f and only i f ( 2 ) = 1 I n t h e gene-
r a l c a s e , l e t I 5 J be t h e I-component of t h e minimal
l e f t almost s p l i t map going o u t from I (assuming,
wi thout l o s s o f g e n e r a l i t y , t h a t I i s indecomposable).
Since f is t h e minimal l e f t almost s p l i t of mod A9:
going o u t from I , l ( 2 ) = R ( J ) + 1. On t h e o t h e r
hand, by t h e l e m m a , [ ( I ) = L ( J ) + 1. [I
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
MERKLEN
f C o r o l l a r y , I f I -+ J i s a morphism o f add 2 - which i s
an 1 - r e ~ r e s e n t a t i v e o f f, thfn $ ( k e r f ) = l(ker 1). - -
P r o p o s i t i o n 2 , s . Let be h e r e d i t a r y and l e t I f J
be a morphism o f add 1 - which i s an i - r e p r e s e n t a t i v e - of f . I f I ' i s a d i r e c t summand o f I such t h a t
PROOF. We can assume t h a t J i s indecomposable and ,
t h e n , t h a t f and f a r e epimorphisms. Le t us proceed
by i n d u c t i o n on l ( I ) , t h e s t a t e m e n t b e i n g vacuous ly
t r u e i n c a s e t h a t $ ( I ) = R ( J ) . W r i t i n g I = I1 @ I '
and , a c c o r d i n g l y , f = ( f l , f 1 ) , we have t h a t fl and
hence f l a r e epimorphisms. If f l i s i n j e c t i v e , f
i s s p l i t e p i c and ~ ( 1 ' ) = & ( k e r f ) , S i n c e k e r f h a s
t o be a d i r e c t summand o f k e r f, it fo l lows from t h e
Coro l l a ry t o Prop . 2 . 4 t h a t k e r f = I t , I f f l i s no t
i n j e c t i v e , f f a c t o r s t h rough I l / k e r f l @ I and t h e
r e s u l t f o l l ows from t h e i n d u c t i o n h y p o t h e s i s ,
i o n 2 , 6 . ( P r e p a r a t i o n lemma) Let I - be h e r e d i -
t a r y and l e t i, g , & be morphisms o f add I/! - - such
t h a t f = p, 2. If f , h a r e I - r e p r e s - e n t a t i v e s o f f , G , - - - - r e s p e c t i v e l y , t h e n t h e r e e x i s t s an 4 - r e p r e s e n t a t i v e - g
o f g such t h a t f = gh, - -
PROOF. We c o n s i d e r f i r s t t h e c a s e when h i s an epimor-
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3159
phism, h = t : I -+ K , whose k e r n e l i s s i m p l e , C l e a r l y ,
i t i s enough t o prove t h e s t a t e m e n t when K i s indecom-
p o s a b l e , I f f = 0, f = 0 and we can t a k e g = 0, - Otherwise , f and f a r e epimorphisms, I f we c o n s i - - d e r t a s a homomorphism between i n j e c t i v e A*-modules, - we s e e t h a t 2 i s o f t h e form
where I1 i s an indecomposable d i r e c t summand o f I
and where tl i s t h e minimal l e f t a lmos t s p l i t map go-
i n g o u t from 11, S ince t h e r e s t r i c t i o n o f f t o I1
cannot be an isomorphism, f f a c t o r s a s f = g ' t . But
t h e n g = , s o t h a t t h e s t a t e m e n t i s proved i n t h i s
c a s e . I n t h e g e n e r a l c a s e , l e t us assume f i r s t t h a t h
i s a s p l i t monomorphismo Then we have e x p r e s s i o n s o f t h e
1 form h = (0) , g = (g1 ,g2) , imply ing t h a t gh = gl ,
gh = f = gl, and o u r problem i s s o l v e d by t a k i n g gl = - - f and g2 = 0, I f h i s n o t s p l i t monic, we have a
f a c t o r i z a t i o n h = h't where t i s a s i n t h e f i r s t - p a r t of t h e p r o o f . Using t h i s , we know t h a t t h e r e e x i s t
morphisrns f ' , h ' such t h a t f = f ' t , h h ' t and
which a r e I - r e p r e s e n t a t i v e s - o f f', h', r e s p e c t i v e l y ,
Then, from f = f't = gh't we deduce t h a t f ' = @' - and t h e p roo f i s completed by induction o n R(I)
( n o t e t h a t , when L ( I ) = L ( K ) , f i s an isomorphism and
I h i s s p l i t mon ic ) , 0
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
3160 MERKLEN
Corollary, Let 2 - be hereditary and let f, g be mor-
phisms from I to J in add 2 . - - If f g are &-representatives, - a necessary and sufficient condition
for the equality f = g is that there exists an auto-
morphism u of 3 such that g = uf $ 5 = I..
Proposition 2.7. Let & be her,e-ditary, The following
properties hold - 1) - If I f 3, J 5 K are morphisms of add I which
are 4-representatives, then gf is an $-re- - presentative,
f 2 ) Let I -+ J be a morphism of add & and let I
= r i , 3 = @ J ~ be decompositions into direct I 3
sums of indecomposable A-modules, Finally, let
f j i : I i J j - be the r~rrespondingfomponents of f. The_n, f is an J-representative if and - only if all the fji's are &-representatives.
PROOF, 1) It is enough to examine the case when f is
split monic and g is epic. Let K 2 Kl @ K2, where
K1 is the image of gf, and let p:K --+ K 2 be the
corresponding projection, We have that pg is an &-re-
presentative and that ~ g ( f (11) = 0 . By Prop. 2 - 5 ,
pg(f(1) = 0 and it follows that gf(I) = K1, showing
that gf is an &-representative. 21 Let li:Ii + I
(respa n :J + J ) be the ith-injection (resp. the j t h 3
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3161
projection) associated to the given decomposition of I
(resp, of J), By l), if f is an 2-representative, - fji = ".fli
J is also an 2-representative, Conversely,
given morphisms fji:Ii + J which are I-representa- j -
tives, let f = 1 ajfjini. Next, let us consider a de- i,j
composition J = J1 83 J2, where J1 is the image of I, and let n be the corresponding projection of J onto
J2. By 11, at jfji~i is an I-representative and, -
since injLijlli = 0, vxjfjini = 0. This means that ~f
= 0, so that f(1) = J1 and f is an J-representa-
tive. 0
Corollary. Let i :I + J be a A-monomorphism with I, J
in add 2 and let us assume that 2 is hereditary, A - - - necessary and sufficient condition for i to be split
is that for every projection T of J onto an indecom-
posable direct summand of J, either ai = 0 or +#0.
3 , The category Comod(add z/Y) when I is hereditary, - - - =
In this section we use the results of section 2 to
obtain a nice description of the category Comod(add I/ -
y ) when I is hereditary. We keep our general nota- - -
tions and assume throughout the section that I is he- - 1
reditary,
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
MERKLEN
We r e c a l l t h a t Comod(add I/Y) - - i s t h e c a t e g o r y f
whose o b j e c t s a r e t h e morphlsms I J o f add I / y - -
and whose morphisms a r e d e f i n e d a s f o l l o w s , I n t h i s ca- f '
t e g o r y , t h e group o f morphisms from - f t o I' > J' i s t h e q u o t i e n t o f t h e group o f p a i r s o f morphisms (a,g) (where 2 i s a morphism from I t o I ' and - B a mor-
phism from J t o J ' ) s a t i s f y i n g f'a = Bf modulo t h e
subgroup o f t h e p a i r s (3,6) such t h a t - n f a c t o r s
t h rough f, i . e , t h e r e e x i s t s a morphism o such t h a t
n = af. The c a t e g o r y Comod(add 1) i s d e f i n e d s i m i l a r - - -
l y and it i s c l e a r t h a t we have an obvious q u o t i e n t
f u n c t o r from Comod(add I) t o Comod(add &/U). - A s we
show n e x t , t h e r e s u l t s a t t h e end o f s e c t i o n 2 imply
t h a t t h i s f u n c t o r has a r i g h t q u a s i - i n v e r s e .
Let - C be the subca t ego ry o f Comod(add 2 ) whose
ob , jec ts a r e d e f i n e d by morphisms f which a r e $ - r ep re -
s e n t a t i v e s and whose morphisms a r e g iven by p a i r s ( a , @ )
such t h a t a and 6 a r e I - r e p r e s e n t a t i v e s , Then, t h e
r e s t r i c t i o n t o C of t h e a b v e mentioned q u o t i e n t func-
t o r i s an e q u i v a l e n c e o f c a t e g o r i e s , S i n c e every o b j e c t
of Comod(add & / v ) - has an & - r e p r e s e n t a t i v e , t h e func -
t o r i s d e n s e , But it i s a l s o f u l l because , g iven a mor-
phism ( g , g ) from t o L' i n Comod(add I / Y ) - - and
choos ing a , f , f ' , i t f o l l o w s from Props . 2 . 6 and 2,7
t h a t t h e r e e x i s t s an I - r e p r e s e n t a t i v e - B o f - B such
t h a t f'a = Bf, It i s a l s o c l e a r from t h e p r e p a r a t i o n
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 31 63
lemma that (r),i) is equivalent to zero if and only if
( ~ ~ 6 ) is equivalent to zero, hence our functor is also
faithful, ending the proof that it is an equivalence of
categories,
Up to isomorphism, any object of Comod(add I/!) is - - f
given by an epimorphism I 2 J, so that it may be re-
presented by an epimorphism I $ J o As a matter of
fact, we would rather represent f by the exact sequen-
ce defined by f : 0 + ker f -+ I f J -+ 0, Accordingly,
a morphism (g,B) from f to f' will be represented - - by a morphism between the corresponding exact sequences,
i,e, by a commutative diagram with exact rows as fol-
lows, where a, B are &-representatives,
Given 2 and B such that f a = Q, our repre- - - sentation procedure determines all objects in the dia-
gram above up to isomorphism, It is easy to see also
that (2,g) is an epimorphism (resp. a monomorphism) of
Comod(add I/y) if and only if 4 is an epimorphism - -
(resp. a monomorphism), Also, if (2,g) is split epic
(resp, monic), then 4 is split epic (resp, manic), The
converse of the first alternative is true when one deals
I with morphisms f which, in some sense, are minimal.
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
MERKLEN 3164
Let us assume t h a t 2 i s such t h a t i t s k e r n e l , i , i s t h e minimal i - enve lope - o f M ( s e e diagram a b o v e ) .
Then we c l a i m t h a t i s s p l i t e p i c i f and o n l y i f
4 i s s p l i t e p i c . I n f a c t , i f M = M1 @ M 2 t h e n I i s
t h e d i r e c t sum I1 @ I p o f t h e minimal 1-envelope of -
M l , 11, and t h e minimal 2-envelope - of M 2 , 12.
T h e r e f o r e , i decomposes i n t h e form il B i2 and f
i n t h e form f l B f 2 . Now, i f $ i s s p l i t e p i c , s o
t h a t M = M' @ M" and 4 = (1,O ) , t h e n we have t h a t
I = I @ I and a has t h e m a t r i x form ( 1 , a " ) . Th i s
e a s i l y i m p l i e s t h a t i s s p l i t e p i c ,
1
P r o p o s i t i o n 3.1. Let M 1 ' be a A-homomorphism
where M i s i n modvA - - and I ' i s i n add 2. -
1) I n o r d e r t h a t i ' be t h e minimal I - enve lope
o f M it i s neces sa ry and s u f f i c i e n t t h a t I '
- M c I " c I ' w i t h I" i n
add I - i m p l i e s I" = - 1'.
be i d i r e c t decomposi t ion o f I ' 2) Let I ' = @ I j - J
i n t o indecomposables and l e t i j :M + I be t h e j -
co r r e spond ing component o f i ' , i ' i s t h e
minimal I -envelope - o f M y t h e n i j f 0 foy -
a 1 1 j; c o n v e r s e l y , i f t h i s c o n d i t i o n i s s a t i s -
f i e d , t h e n i ' i s an ;-envelope o f M .
PROOF. 1) ~~t M .& 1 be t h e minimal &-envelope of M .
Then we have a commutative d iagram w i t h e x a c t rows a s
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIYES 3165
fo l lows , where f is t h e cokerne l of i and f ' i s the
cokernel o f i'.
I f I ' i s minimal, a is s u r j e c t i v e and so i s 6.
Since i i s a minimal morphism, t h e gene-
r a l remarks preceding t h e p r o p o s i t i o n show t h a t a i s
an isomorphism. The converse i s obvious by d e f i n i t i o n
o f minimal envelope. 2 ) L e t us suppose t h a t , f o r some j,
i is 0 modulo 1. Then, i f a c t o r s through some j j
module P i n and, s i n c e T(P) i s conta ined i n - r a d P , we see t h a t I T is no t minimal. Conversely, i f
a l l t h e components i a r e d i f f e r e n t from O modulo y, j -
it i s easy t o conclude, from t h e c o r o l l a r y to Prop. 2 . 7 ,
t h a t t h e minimal 2-envelope - o f M i s a d i r e c t summand
of 1'.
Propos i t ion 3 . 2 . Given an indecomposable module M t h a t
i s n o t i n , let i be t h e minimal &-envelope of M
and l e t f be t h e cokerne l o f i. Then t h e e x a c t se -
quence de f ined by these two ma-ps i s a l s o exac t modulo y. -
i f O + M - - - . , I - J + O
PROOF. We know from [ 41 t h a t i f P is i n Y, - t h e n -
T(P) i s i n add 5 . - It fol lows t h a t M i s con ta ined i n
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
3166 MERKLEN
some module o f add I - and , hence , i i s monic, On t h e
o t h e r hand , we know t h a t f i s e p i c and we can show
t h a t it i s i n f a c t t h e c o k e r n e l of Le t us c o n s i -
d e r t h e f o l l o w i n g commutative d iagram where j i s t h e
y-envelope o f M and where X i s t h e pushou t o f j - and i ,
t h e o t h e r hand , s i n c e j f a c t o r s t h rough i , t h e up-
p e r row s p l i t s and t h e r e f o r e g = f o We show n e x t t h a t , -
d given N ---L I, i f 3 = 0, f a c t o r s t h rough i. We
can assume t h a t $ # 0 and t h a t N i s i n modVA. Let - -
N I1 be t h e I - enve lope o f N and l e t f l=cok i - 1'
Then f a c t o r s a s a and , i f a i s an I - r e p r e -
s e n t a t i v e o f a ' , t h e r e i s an & - r e p r e s e n t a t i v e B such
t h a t f a = @fig ( I n f a c t , (&)il = 0 i m p l i e s t h a t fa - f a c t o r s t h rough f l and w e g e t B u s i n g t h e p r e p a r a - - t i o n lemma.) Then, f a i l = 0 i m p l i e s t h a t a i l f a c t o r s
t h rough i: a = i F i n a l l y , 9 = s'il = - a i l = - +'i - -
a s we wanted t o prove . To comple te t h e p r o o f , it i s ne-
c e s s a r y t o show t h a t i i s monic, For t h i s we show
f i r s t an a u x i l i a r y f a c t ,
Let I ' be a module i n add 2 - which i s c o n t a i n e d
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3167
i n I and such t h a t t h e i n c l u s i o n j o f I ' i n t o I
i s z e r o modulo y o Then, we c l a i m , I ' i s c o n t a i n e d i n - M, I n o r d e r t o prove t h i s we show t h a t M # M + I ' i m -
p l i e s j # 0. Let us c o n s i d e r t h e f o l l o w i n g d iagram - w i t h obvious maps,
I t fo l lows t h a t T # 0 , because fi' = 0 i m p l i e s a - f a c t o r i z a t i o n o f t h e form L' = i b and , s i n c e i = i ' a , - i t ( l - 0 6 ) = 0 ; b u t , s i n c e M c anno t be a d i r e c t summand
o f M + I 1 , 1-06 i s a n automorphism o f M + I 1 , imply ing
t h e c o n t r a d i c t i o n i t = 0, Hence, l e t us s u b s t i t u t e t h e - t o p row o f t h e d iagram above by t h e row 0 + I ' I -
4 cok j , and a by t h e i n c l u s i o n CY o f I' i n t o
S i n c e i s e p i c and t h i s , t oge -
t h e r w i t h t h e f a c t t h a t # 0 , i m p l i e s j # 0. - Now we u s e t h i s a u x i l i a r y f a c t t o prove t h a t i i s -
manic. We remark f i r s t t h a t , g iven a d iagram M I 2 J
where i i s t h e &-envelope o f M , where J i s i n
add I - and where a i s an & - r e p r e s e n t a t i v e and M i s
i n modVA, t h e n ai = 0 i m p l i e s a i = 0, I n f a c t , i f - -
f = cok A , t h e g i v e n r e l a t i o n i m p l i e s a f a c t o r i z a t i o n - cu_ = af where, by t h e p r e p a r a t i o n lemma, a may be cho-
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
31 68 MERKLEW
s e n s o t h a t a = o f , hence ai = a f i = 0 . On t h e o t h e r
f hand, we r e c a l l t h a t i f I + N i s a A-homomorphism
where I is i n add 4 and where N h a s no d i r e c t sum-
mand i n add ;, - t h e n f = O o F o r p rov ing t h a t i is
m o n i c w e can assume t h a t M does not be long t o I. - Let
g us be given a map N - M w i t h g f 0 such t h a t i g = - - k
0, We can assume t h a t N i s i n modVA. C a l l i n g N +- K - - t h e &-envelope - of N , we have t h a t i g f a c t o r s i n t h e
form a k . Let I1 be t h e image of 2 and l e t I = I1
al 8 I?, a = ( ) be t h e co r re spond ing decomposi t ions o f a 2
I and a . Then, al i s a n $ - r e p r e s e n t a t i v e and t2 i 1 = 0, I f ( i 1 i s t h e co r re spond ing decomposition of
2 i, t h e r e l a t i o n = ilg = 0 i m p l i e s a lk = 0 and ,
hence, i l g = 0, On t h e o t h e r hand, by t h e a u x i l i a r y
f a c t , a 2 ( K ) i s a module i n add I_ - which i s con ta ined
i n M. There fo re , t h e i n c l u s i o n o f a 2 ( K ) i n t o M i s
0 modulo y. - But, from what we have a l r e a d y s e e n , g
f a c t o r s t h rough t h i s map and we g e t t h e d e s i r e d conclu-
s i o n t h a t g = 0 . - D
Coro l l a ry . L e t M N b e a morphism o f modVA and l e t - -
us assume t h a t ; i s hered- i ta ry , - i s a n epimorphism, i f and o n l y
b ) i s a n epimorphism i f and o n l y i f is a n - - epimorphism,
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3169
PROOF, P a r t a ) i s e s s e n t i a l l y ( 3 ) o f Prop , 2-1, which i s
t r u e i n o u r c a s e ( s e e remark f o l l o w i n g Def, I), The
proof t h a t 4 e p i c i m p l i e s e p i c i s t h e same a s i n - t h e proof o f Prop. 2 , 3 , F i n a l l y , t h e proof t h a t i f 4
i s a n epimorphism t h e n 4 i s a n epimorphism i s an ea- - sy consequence o f Prop , 3 , 2 and i s l e f t a s an e x e r c i s e ,
An i n t e r e s t i n g consequence o f t h i s C o r o l l a r y is
t h a t , i f t h e r e i s an e q u i v a l e n c e moduLo p r e p r o j e c t i v e s
up t o t h e l e v e l n between A and a h e r e d i t a r y a i g e -
b r a A , i f - i s h e r e d i t a r y t h e e q u i v a l e n c e f u n c t o r
c a r r i e s P n + l ( A ) o n t o En+ l A Hence, by i n d u c t i o n ,
t h e r e i s such an e q u i v a l e n c e up t o t h e l e v e l m , f o r
any m g r e a t e r t h a n n o 'This poses t h e i n t e r e s t i n g
q u e s t i o n of a n a l y s i n g p r o p e r t i e s o f t h e minimum o f t h e
numbers m w i t h t h i s p r o p e r t y ,
I n o r d e r t o s t a t e t h e f o l l o w i n g p r o p o s i t i o n , which
i s one of t h e main r e s u l t s of t h i s s e c t i o n , l e t us i n -
t r o d u c e some more t e rmino logy and n o t a t i o n , As it was
'po in ted o u t b e f o r e Prop , 2 , 4 , we can i d e n t i f y Comod(
add I/?) w i t h mod A$c , where A* i s a h e r e d i t a r y a l - - - g e b r a , and we saw a l s o , a t t h e beg inn ing o f t h i s sec-
t i o n , t h a t mod A:\ i s e q u i v a l e n t t o t h e subca t ego ry
of Comod(add E) d e f i n e d by morphisms f t h a t a r e 4- - - r e p r e s e n t a t i v e s and c o n s i d e r i n g a s morphisms o n l y t h o s e
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
31 70 MERKLEN
( a , B ) such t h a t cl and a r e 2 - r e p r e s e n t a t i v e s . -
The f u n c t o r H which g i v e s t h i s e q u i v a l e n c e i s t h e
r e s t r i c t i o n t o 2 of t h e q u o t i e n t f u n c t o r Comod(add I) -
Comod(add I/!), - - We w i l l d e n o t e by C/! t h e f u l l
subca t ego ry o f 2 d e f i n e d by epimorphisms f whose
k e r n e l i s t h e I - enve lope - o f a module i n modVA ( i n - -
o t h e r words, f co r r e sponds t o an e x a c t sequence o f i f
t h e form U + M + I + J + 0 where M h a s no indecom-
posab le summands i n V - and where i i s t h e minimal I- - - envelope o f M ) , Le t us i n t r o d u c e a l s o t h e c a t e g o r y y: t h e f u l l subca t ego ry o f mod A * c o r r e s p o n d i n g t o ob-
j e c t s - f o f Comod(add I/!) r e p r e s e n t e d by e x a c t s e - - - quences a s above where I i s i n I and i = 0, F i n a l - - l y , we w i l l d eno te by y9: t h e f u l l subca t ego ry of -
mod A:': d e f i n e d by modules which have a n epimorphism on-
t o an o b j e c t o f I t i s ea sy t o s e e t h a t !$: i s - 0
c l o s e d f o r submodules and , hence , t h a t mody,A5 i s -
e q u i v a l e n t t o mod A"/!*, - The f o l l o w i n g p r o p o s i t i o n i s
now a n easy consequence o f Prop , 3 , 2 ,
Proposition - -.- 3 , 3 , The f o l l o w i n g c a t e g o r i e s a r e equiva-
l e n t , when I i s h e r e d i t a r y , -
mod A 7'
mod A* V" -
PROOF, If H/y i s t h e r e s t r i c t i o n o f H t o C/V, - - - t h e n t h e image o f H / y i s t h e f u l l subca t ego ry o f -
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 31 7 1
mod A * g i v e n by t h e morphisms f which, w i t h o u r no-
t a t i o n s , a r e such t h a t i i s t h e minimal :-envelope -
of a module i n modVA, By P ropa 3 , 1 , t h i s happens i f - -
i and o n l y i f none o f t h e components o f M -+ I = @ I
j j (Ij indecomposable) i s 0 modulo y o On t h e o t h e r hand, - i f one o f t h e s e components, i were 0 modulo y , we
j -
would have an epimorphism
where, by d e f i n i t i o n , f . i s i n V k I0
Th i s shows t h a t -1
i f - f i s no t i n t h e image o f H / y , - t h e n f i s i n y;'2, -
Conver se ly , i f - f i s i n V " , - ( u s i n g o u r r e p r e s e n t a t i o n - by s h o r t e x a c t s equences ) we have a n epimorphism
w i t h f ' i n V*, I f it were t h a t a l l components i - = O j
o f i a r e d i f f e r e n t from O modulo P , - u s i n g Prop ,
3 , l and Prop , 3 , 2 , t h e f a c t t h a t a> z 0 would i m -
p l y t h a t (g,g) = 0, a c o n t r a d i c t i o n , Thus we have
t h a t H/Y - d e f i n e s a n e q u i v a l e n c e from C / y - t o r n ~ d ~ , f l ; ~ , - -
To comple te t h e p r o o f , l e t us c o n s i d e r t h e f o l l o w i n g
f u n c t o r from modVA t o C / y . Given an indecomposable - -
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
3172 MERKLEN
M i n mod,,ii, we a s s o c i a t e t o it t h e c o k e r n e l f o f - - $ f
t h e I -envelope - of M, Given a morphism M 7 M ' o f t h e
ca t ego ry modVA / y , we can b u i l t a commutative d i a - - - - gram w i t h e x a c t rows a s f o l l o w s o
Then, we choose & - r e p r e s e n t a t i v e s a , B , & of a ' , & ' , - r e s p e c t i v e l y such t h a t f 'a = p f , By d e f i n i t i o n , o u r
f u n c t o r a s s o c i a t e s w i t h ( a , @ ) . Le t u s remark t h a t
( a , B ) d e f m e s , by pas sage t o t h e k e r n e l s , a morphism
' such t h a t a i = i f $ Then, ( 2 - ' ) = 0 i m p l i e s
t h a t - $ = 2 and hence (%,&I d e f i n e s t h e same mor-
pF.ism a s (%' , D l ) I t i s easy t o s e e now t h a t t h i s func -
t o r i s f u l l and f a i t h f u l ,
According t o t h i s p r o p o s i t i o n , when I i s h e r e d i t a - - r y , a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r ii t o b e
e q u i v a l e n t , modulo p r e p r o j e c t i v e s up t o t h e Level n , t o
a h e r e d i t a r y a l g e b r a i s t h a t c o i n c i d e s w i t h
add pn(ii9:). The r e s t o f t h i s s e c t i o n i s e s s e n t i a l l y de- - vo ted t o g i v e some i n f o r m a t i o n abou t t h e r e l a t i o n be t -
ween t h e s e two c a t e g o r i e s , We w i l l d e n o t e by t h e .. f u l l subca t ego ry o f d e f i n e d by t h e indecomposable
modules P such t h a t P i s c o n t a i n e d i n some module
of add I - ( t h e s e a r e p r e c i s e l y t h e modules i n en ( i i ) -
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3173
whose 2-envelope - i s a monomorphism~, It should be re -
mbkeci t h a t P e q u a l s i f and on ly i f t h e r e a r e no - - i n j e c t i v e modules i n y o To each P i n w e asso- - -
f~ c i a t e t h e exact sequence 0 + P I --t J + 0, where i
i s t h e minimal I-envelope of P, -
P r o p o s i t i o n 3 , 4 . Given P in P , w i t h t h e n o t a t i o n s a- - bove (and assuming I - i s h e r e d i t a r y ) , if P is i n $,(A> t hen f p i s i n E m , < A a , , f o r some m t t h a t - - i s l e s s than o r equ g = y, m 1 = m. - -
PROOF, We proceed by i n d u c t i o n on m , i f m = 0, l e t
P be a n indecomposable p r o j e c t i v e which l i e s i n P - and
l e t (cx-,~) be an epimorphisrn onto f Using o u r r e - -P "
p r e s e n t a t i o n by exact sequences, we have a commutative
diagram wi th exact rows o f t h e fo l lowing form,
We observe t h a t i n a diagram of t h i s form, because i
i s a minimal 2-envelope, - i f i s a s p l i t epimorphism,
then (g ,g ) i s a s p l i t epimorphism. This i s a comple-
ment t o t h e remark preceding Prop, 3 , l t h a t fo l lows ea-
s y l y from t h e f a c t t h a t i f P t ' ~ ' i s such t h a t 4 + t =
1, then +' may be extended t o a morphism (a- ' ,gl)
from f p t o f' such t h a t ( ~ , ~ ) ( ~ ' , & ' > = 1. Hence,
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
3174 MERKLEN
s i n c e i s a s p l i t epimorphism, ((yl,g) i s s p l i t and
t h i s p roves t h a t f p i s p r o j e c t i v e a s a A"module, For
t h e i n d u c t i o n s t e p , t h e same argument i s used s t a r t i n g
from t h e same 'd iagram above where , now, it i s assumed
t h a t P i s i n E m ( A ) , t h a t (g,!) i s a n epimorphlsm
and P h a s no indecomposable summands i n pm-l ( A ) ,
The consequence i s t h a t ip l i e s i n e r n ( h * ) , - comple-
t i n g t h e proof o f t h e f i r s t p a r t o f o u r p r o p o s i t i o n , I n
t h e c a s e t h a t E = y , we can assume t h a t P' i s a - - cove r of P a n d , p roceed ing by i n d u c t i o n , t h a t it i s
i n add l?m-l(AA). Hence, i f i s o b t a i n e d s t a r -
t i n g from , w e g e t a non s p l i t epimorphism showing
t h a t ip i s n o t i n ( A ~ ) -
PROOF, We show f i r s t t h a t V" i s c o n t a i n e d i n - - add en() ,") o r , a s it i s enough, t h a t h a s t h i s - p r o p e r t y , Let f d e f i n e a n o b j e c t of Y: r e p r e s e n t e d - b y t h e e x a c t sequence 0 + M i I 5 J -+ 0, C l e a r l y ,
s i n c e A * i s h e r e d i t a r y , we c a n assume, w i t h o u t i o s s o f
g e n e r a l i t y , t h a t M i s maximal i n I such t h a t = 0'
Since i f a c t o r s t h r o u g h some P i n Y i n t h e form -
M 2 P I, it i s e a s y t o s e e t h a t I i s a component o f
i t t h e I - enve lope o f P, P + 1 . I n f a c t , c o n s i d e r i n g -
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3175
t h e f o l l o w i n g d iagram, where a i s such t h a t r = a i r ,
we s e e t h a t i s a n isomorphism o f Comod(add I/!), - -
A s a consequence , d e f i n e s a non-zero morphism from
f t o cok i t and t h i s , by Prop. 3 , 4 , i s i n add pn (Ae) , - -
I n o r d e r t o comple te t h e p r o o f , we show now t h a t indecom-
p o s a b l e A-modules o u t s i d e of y* do not be long t o
add pn(Ass) . Such a module may be r e p r e s e n t e d by an e x a c t
sequence o f t h e form 0 + M $ I 5 J -+ 0 w i t h M i nde -
composable and where i i s t h e minimal I - enve lope o f - M ( s e e Prop , 3 , 3 ) , Then, u s i n g a P (A)-cover o f M y - n
P + M y it i s p o s s i b l e t o d e f i n e a non s p l i t epimor-
phism from a module i n add E n ( A ) o n t o 2 , Hence, f - i s n o t i n add p n ( h 5 ) ,
Remark, I n t h e g e n e r a l c a s e , a n e c e s s a r y and s u f f i -
c i e n t c o n d i t i o n f o r y A - t o be composed o f p r e p r o j e c t i v e
modules i s t h a t a l l e x a c t sequences 0 + M 4 I 5 J + 0
w i t h M and I i n I, and M maximal under t h e s e - c o n d i t i o n s , r e p r e s e n t p r e p r o j e c t i v e A%-modules. I n
f a c t , w i t h t h e n o t a t i o n s o f t h e proof above , M = T T ( P )
i s i n T o - I t would b e i n t e r e s t i n g t o f i n d c o n d i t i o n s
f o r t h i s and , i n p a r t i c u l a r , t o c h a r a c t e r i z e when y7': -
i s of t h e form add pm(AA) f o r some m ,
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
P r o p o s i t i o n 306., Let 1 - be h e r e d i t a r y and l e t us assume
that component o f I -
If a l l indecomposable modules i n y* - a r e p r e p r o j e e t i v e ,
t h e n every I I n 3 is a q u o t i e n t o f a n i n j e c t i v e A-
module be long ing t o f o r o f a module of t h e form T ( P ) ,
where P i s a n indecomposable i n j e c t i v e A-module be-
l o n g i n g t o I n o t h e r words, t h e s o u r c e s o f 4 = i n j e c t i v e o r a r e o f t h e form T(P) , P indecomposable
i n j e c t i v e i n y. -
PROOF, Proceeding by c o n t r a d i c t i o n , l e t L be a n e l e -
ment o f I for which t h e s t a t e m e n t i s f a l s e and l e t u s - choose L o f maximal p o s s i b l e l e n g t h , I f I. i s t h e
i n j e c t i v e envelope o f L , t h e r e e x i s t s a module I',
belonging t o I, - such t h a t I ' l i e s i n between L and
I. and does no t c o i n c i d e w i t h L . L e t us choose I '
minimal under t h o s e c o n d i t i o n s and let us c o n s i d e r t h e
e x a c t sequence d e f i n e d by t h e i n c l u s i o n o f L i n t o I t ,
0 + L 4' I t 5' J q + 0 . By t h e h y p o t h e s i s , each r i n g
component o f A " i s of i n f i n i t e r e p r e s e n t a t i o n t y p e
and hence, s i n c e 0 + L 1 L + 0 -+ 0 r e p r e s e n t s an in -
j e c t i v e A:%-module, t h e r e is a n i n f i n i t y o f modules i n
modVgaA*, w i t h no indecomposable components i n common, - - having t h i s i n j e c t i v e a s image under a n epimorphism.
Those modules c a n be r e p r e s e n t e d by e x a c t sequences o f
t h e form 0 + M + I1 fL J1 -+ 0, s o t h a t t h e r e i s a n
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 3177
infinity of A-modules M, with no indecomposable sum-
mands in common, for which there are epimorphisms of
@ the form M 4 L, Let I' - be the subquiver of de-
fined by the modules which satisfy our statement (notice
that 1' is one of them), For each of the above mentio-
ned modules M, let i be the minimal if-envelope of - M , Using the epimorphism @ we obtain a commutative
diagram as follows,
By the choice of L, a (I) cannot be equal to L and,
therefore, has to be equal to I f , Then, for each M,
we have a non split epimorphism (k,g) onto f' , con-
tradiction to the fact that f' is preprojective in
mod A*,
Propositjon 3 , 7 , Let - be hereditary and let us as-
sume that there are no injective A-modules in y o Then - - I - = Im(&o)o r words the sources of 2 are in- - -
j ective A-modules - - ,
PROOF. We proceed in the same way as in the proof of
Prop. 3 , 6 up to the choice of I' , Let M L be the
Cn(A)-cover of L and let i be the injective envelope
of M. As in the proof of Prop, 3 , 6 , we can construct
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
3178 NERKLEN
now a non s p l i t automorphism o f f o n t o f' and t h i s
g i v e s a c o n t r a d i c t i o n t o P r o p , 3 , 5 ,
4 . P roof of t h e main t heo rem, An example.
Le t u s assume t h a t I i s h e r e d i t a r y and t h a t t h e r e
a r e no i n j e c t i v e A-modules i n y . Then P = and it - - -
f o l l o w s f rom P r o p , 3 , 3 and P r o p o 3 . 5 t h a t A i s equ iva -
l e n t , modulo p r e p r o j e c t i v e s up t o t h e l e v e l n , t o t h e
h e r e d i t a r y a l g e b r a A " , A l s o , by Prop . 2 . 1 , ( 3 ) , A s a -
t i s f i e s t h e p r o p e r t y s t a t e d i n t h e t heo rem ( c f , remark
f o l l o w i n g D e f , l ) , F i n a l l y , P r o p , 3 , 7 s a y s t h a t I = I m ( ~ o ) o -
Le t u s assume now t h a t A h a s t h e p r o p e r t y s t a t e d
i n t h e t heo rem and t h a t it i s e q u i v a l e n t , modulo p r e p r o -
j e c t i v e s up t o t h e l e v e l n , t o a h e r e d i t a r y a l g e b r a
w i t h no r i n g components of f i n i t e r e p r e s e n t a t i o n t y p e ,
Then Prop . 2 . 2 s a y s t h a t 2 - i s h e r e d i t a r y ,
Now w e go t o t h e proof t h a t s t a t e m e n t s 1) and 2 )
o f t h e main t heo rem a r e e q u i v a l e n t , L e t u s assume t h a t
5 - i s h e r e d i t a r y , Then 6 - = I m ( I o ) and we have Zb ) ,
On t h e o t h e r hand , 2a) i s a n e a s y consequence o f P r o p ,
3 .2 . Conve r se ly , l e t us assume t h a P 2a) and 2b) a r e
s a t i s f i e d , I f I i s i n I, - e i t h e r I i s i n j e c t i v e o r
t h e r e i s a n i r r e d u c i b l e map I -+ P w i t h P i n !j. - If
i t s image were n o t c o n t a i n e d i n r a d P, we would e a s i l y
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODULO PREPROJECTIVES 31 79
o b t a i n a c o n t r a d i c t i o n t o 2 a ) - Hence I i s a d i r e c t
summand o f r a d P and , by 2 b ) , it i s i n I m ( ~ o ) o I t
f o l l o w s from 2a) t h a t i r r e d u c i b l e maps between elem-
e n t s o f 2 a r e epimorphisms. Then, g i v e n I i n t h e - l e f t boundary of which i s n o t !-simple, it i s e a s y - - t o s e e t h a t t h e r e a r e i r r e d u c i b l e maps o f t h e form
I + J and J $ TrD I w i t h g a monomorphism. Hence,
I c o n t a i n s t h e l e f t boundary o f and , by what we - - have a l r e a d y e s t a b l i s h e d , 2 i s open t o t h e r i g h t i n - r A \ g n ( ~ ) and does n o t c o n t a i n o r i e n t e d c y c l e s . F i n a l -
l y , u s i n g 2a) once a g a i n , it f o l l o w s t h a t P rop , 2 . 1 ,
(1) i s s a t i s f i e d and I i s h e r e d i t a r y by t h e remark - f o l l o w i n g Def, 1,
EXAMPLE, Le t k b e a f i e l d and T = ki XI / (x3), We de-
n o t e by I t h e r a d i c a l o f T and by S i t s s i m p l e
module T / I , L e t u s c o n s i d e r t h e f o l l o w i n g f i n i t e d i -
mens iona l a l g e b r a o v e r k.
We s e e t h a t A c anno t b e e q u i v a l e n t t o a h e r e d i t a r y a l -
g e b r a modulo p r o j e c t i v e s b e c a u s e t h e non-simple indecom-
p o s a b l e p r o j e c t i v e , P L Y h a s a submodule i somorph ic t o
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
31 80 MEKKLEN
I which i s n e i t h e r p r o j e c t i v e n o r s imp le , Let us d e n o t e
S1, S 2 , P1, P2 t h e s imp le A-modules and t h e i r p r o j e c -
t i v e c o v e r s , r e s p e c t i v e l y , and l e t 11, I 2 b e t h e i n -
j e c t i v e enve lopes of Sly S 2 , r e s p e c t i v e l y o We c a n ea-
s i l y compute t h e f o l l o w i n g p o r t i o n of t h e Auslander-Rei-
t e n q u i v e r o f A .
Hence, f o r n = l , i s g e n e r a t e d by Ply P2, Q1 and Q 2 -
and I c o n s i s t s of I , I , I and S1, A s a conse- -
quence, A i s e q u i v a l e n t , modulo p r e p r o j e c t i v e s up t o
t h e l e v e l 1, t o A*, t h e p a t h a l g e b r a o f t h e q u i v e r
ACKNOWLEDGEMENT
P a r t o f t h i s work was done w h i l e t h e a u t h o r w a s v i s i -
t i n g t h e Mathematische I n s t i t u t d e r U n i v e r s i t a t , Zi ir ich
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13
INJECTIVE SUBQUIVERS AND EQUIVALENCE MODLnO PREPROJECTIVES 3181
w i t h p a r t i a l suppor t by t h e ~ u n d a g z o de Amparo Pesquisa
do Es tad0 de S ~ Q Paulo (FAPESP 1, Braz i l .
REFERENCES
111 - M. Auslander and I . Re i t en , Stable equivalence of
a r t i n a l g e b r a s , S p r i n ~ e r L. N. M. 353 (1972) 8-71.
[ 2 l - M, Auslander and I. Re i t en , Represen ta t ion theory
o f a r t i n a l g e b r a s , 111, Comrn. Algebra 3 (1975)
239-293.
1 3 1 - M. Auslander and S, 0. Smald, P r e p r o j e c t i v e mod-
u l e s o v e r a r t i n a l g e b r a s , J.Algebra 66 t19801 61-
122.
141 - H. Merklen, A r t i n a l g e b r a s which a r e e q u i v a l e n t t o
a h e r e d i t a r y a l g e b r a modulo p r e p r o j e c t i v e s , Sprin-
g e r L, M, M. 1 7 7 7 (19863 232-255.
Received: Juna 1989
Revised: January 1990
Dow
nloa
ded
by [
Uni
vers
itaet
s un
d L
ande
sbib
lioth
ek]
at 1
7:47
03
Sept
embe
r 20
13